Exact controllability of the suspension bridge model proposed by Lazer and McKenna

In this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by Lazer and McKenna in [A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537–578]: wtt + cwt + dwxxxx + kw+ = p(t, x)+ u(t, x) +f (t,w,u(t,x)), 00, c > 0, k > 0, the distributed control u ∈ L2(0, t1;L2(0, 1)), p :R × [0, 1]→R is continuous and bounded, and the non-linear term f : [0, t1]×R×R→R is a continuous function on t and globally Lipschitz in the other variables, i.e., there exists a constant l > 0 such that for all x1, x2,u1,u2 ∈ R we have f (t,x2,u2) −f (t,x1,u1) l x2 − x1 + u2 −u1 , t∈ [0, t1]. To this end, we prove that the linear part of the system is exactly controllable on [0, t1]. Then, we prove that the non-linear system is exactly controllable on [0, t1] for t1 small enough. That is to say, the controllability of the linear system is preserved under the non-linear perturbation −kw+ + p(t, x)+ f (t,w,u(t,x)).  2005 Published by Elsevier Inc